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A hyperbolic toral automorphism is a toral automorphism $T_{A}x=Ax(\mathrm{mod}{\mathbb{Z}}^d)$ whose matrix has no eigenvalue on the unit circle. Here $A\in \mathrm{GL}(d,\mathbb{Z})$, so its determinant is $\pm 1$. Invertibility merely over the real numbers would not suffice: multiplication by two on a circle is a many-to-one endomorphism, not an automorphism.

The real stable and unstable subspaces come from the generalized eigenspaces with eigenvalue modulus less than and greater than one. They give the same splitting at every point of the real torus. A Jordan block can add a polynomial factor to the powers of its eigenvalue; choosing a slightly weaker exponential rate absorbs that factor into a constant. Thus contraction forward on the stable subspace and backward on the unstable subspace is uniform over the whole torus, proving that the map is an Anosov diffeomorphism.

For the cat map, $A=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3\pm \sqrt{5})/2$. One is inside and the other outside the unit circle, and their product is one. The toral automorphism mixing criterion applies as well, since a root of unity would lie on that circle.

Use normalized Haar measure $m$ on the real torus. The integer matrix $A$ and its integer inverse induce a toral automorphism; its determinant one preserves $m$. For $k\in {\mathbb{Z}}^n$, let $e_k(x)=e^{2\pi ik\cdot x}$. These characters of a real torus are an orthonormal basis, andSuppose no eigenvalue of $A$ is a root of unity. For $k\ne 0$, the frequency $(A^T{)}^{j}k$ can equal a fixed $\ell$ at most once. Otherwise two such times would give $(A^T{)}^{q}k=k$ for some $q>0$, forcing $A^q-I$ to be singular and an eigenvalue of $A$ to have $q$th power one. Hence the displayed correlation eventually vanishes whenever a nonconstant character is involved. Constant characters give the product of their means.

Linearity gives mixing for finite Fourier series. To pass to arbitrary $f,g\in L^2(m)$, approximate them by trigonometric polynomials. Composition with $T_A^j$ is an isometry of $L^2(m)$, so the Cauchy-Schwarz inequality bounds the approximation errors uniformly in $j$. Let those errors tend to zero after taking the large-$j$ limit. Thus $\int (f\circ T_A^j)\overline{g}dm\to \int fdm\int \overline{g}dm$, which is the required mixing measure-preserving transformation property.

Conversely, if an eigenvalue is a root of unity, choose $q>0$ with $\det [(A^T{)}^q-I]=0$. This is an integer matrix, so its nullspace has a nonzero rational vector; clearing denominators gives $k\in {\mathbb{Z}}^n\setminus \{0\}$ with $(A^T{)}^{q}k=k$. The nonconstant character $e_k$ has mean zero, butfor every $j$. It cannot satisfy mixing. ThereforeHyperbolic toral automorphisms are mixing, but hyperbolicity is not necessary: this proof excludes periodic integer frequencies, not every eigenvalue of modulus one.

An integer matrix $A$ with determinant $\pm 1$ defines a toral automorphismon the real torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$. Integer entries make the map independent of the chosen representative. The inverse matrix also has integer entries, so the transformation is invertible. It preserves normalized Haar measure.

The map is a mixing measure-preserving transformation exactly when no eigenvalue of $A$ is a root of unity. This can be proved with characters of a real torus $e_k(x)=e^{2\pi ik\cdot x}$, since $e_k\circ T_A=e_{A^{T}k}$. Orthogonality givesIf no eigenvalue is a root of unity, a nonzero integer frequency cannot repeat: a repetition would give $((A^T{)}^m-I)k=0$ for some $m>0$. It therefore eventually leaves every finite set of frequencies. The displayed correlations prove mixing first for trigonometric polynomials, then for square-integrable functions by approximation.

Conversely, a root-of-unity eigenvalue makes $(A^T{)}^m-I$ singular for some $m$. As an integer matrix it has a nonzero rational kernel vector, which can be scaled to a nonzero integer vector $k$. The nonconstant character $e_k$ is then invariant under $T_A^m$, obstructing mixing. The cat map is a mixing example, while the identity matrix gives an invertible measure-preserving map that is not mixing.